Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [Figure 3.1]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

1. [M1] For resolving forces in any one direction
2. [A1] E.g. X=18+12 sin 60°-8 sin 30° 14 + 6√3 One correct equation or expression
3. [A1] E.g. Y = 8 cos 30°+ 12 cos 60° 6+4√3 Second correct equation or expression (X and Y may denote components of resultant of given 3 forces or may be components of the fourth force that would produce equilibrium)
4. [M1] [(14+ 6√3)² + (6+4√3)²] or [tan⁻¹ (6 + 4√3)/( 14 + 6√3)] Use of Pythagoras or appropriate trig to find magnitude or angle
5. [A1] Magnitude is 27.6 (N) Not for resultant
6. [A1] Direction is 27.9° below ‘negative x-axis' Not for 27.9° only; direction must be clearly specified